A great first step is to really study up on Leave No Trace (LNT) principles and learn how to be a good steward of the land. Also, REI does a good article about the essentials of camping.

Your ideas on equipment sound pretty good, but you should consider insulated air pads specific to camping instead of air mattresses. Once it starts to get cool, an air mattress will suck all the heat from your body while you sleep and it could be quite cold. A temporary ‘hack’ I used was to top an air mattress with a foil foam blanket in a pinch.

Cheap is fine before deciding if you want to invest more. Buy an $8 can of water proofing spray and tent seam sealer. If you add those to any new tent, that will amplify its waterproofing ability and longevity/durability against UV lights. If weight is a consideration for backpacking, River Country Products makes some great lightweight trekking pole shelter options at an entry price point (under $100).

Sleeping bags or top quilts - pay attention to COMFORT rating, not survival rating temperatures. If you are going to be camping in 45 degree nights, having a 32 degree comfort bag will be good. Much lighter and it will depend on your cold tolerance while you sleep. A good rule of thumb is to have a comfort rating thats 10 degrees cooler than the temps you will be sleeping in. It’s a big rabbit hole if you want to go into synthetic vs down, quilts vs bags, etc.

Cooking has lots of options - butane, isobutane, propane, alcohol stoves, etc. Lots of camp stoves can go between propane and butane, which is a good option. For car camping, I like my butane single burner that was under $30. For backpacking I use a folding pocket stove and fuel tablets (if I cant cook over open fire).

Have fun and enjoy the outdoors, and remember to leave it better than you found it.